ALIBABA 2021 Problem 4

Let $(\Omega, \mathcal{A},\mathbb{P})$ be a standard probability space, and $\mathcal{X}$ be the set of all bounded random variables. For $t>0$ , defined the mapping $R_t$ by $$R_t(X)=t\log \mathbb{E}[\exp(X/t)], X \in \mathcal{X},$$ and for $t \in (0,1)$ define the mapping $Q_t$ by $$Q_t(X)=\inf\{x \in \mathbb{R}: \mathbb{P}(X>x) \le t\}, X \in \mathcal{X}.$$ For two mappings $f,g:\mathcal{X} \to [-\infty,\infty)$ , defined the operator $\square$ by $$f\square g(X)=\inf\{f(Y)+g(X-Y): Y \in \mathcal{X}\}, X \in \mathcal{X}.$$ (a) Show that, for $t,s>0$ , $$R_t \square R_s=R_{t+s}.$$ (b) Show that, for $t,s>0$ with $t+s<1$ , $$Q_t \square Q_s=Q_{t+s}.$$